New MTA chief Jay Walder wants to improve the state-run authority’s non existent credibility. The way to start should be to launch a full-on, public investigation of the MTA’s strange relations with the Transport Workers Union and its political allies during last year’s contract negotiations.

Such a probe would be an invaluable case study in how, when it comes to labor deals, secrets and lies are not the taxpayers’ friends.

The MTA, we now know, spent a year of contract talks making wildly changing offers that ended with an abrupt declaration of an impasse. A panel of arbitrators decided the deal in the summer: raises totaling 11.3 percent over three years, plus a giveback to the union on health care. (The MTA is now appealing the deal.)

Arbitration likely was a ruse, although we don’t know for sure. We can guess that neither Gov. Paterson nor the MTA thought that awarding huge raises would fly publicly, especially when the MTA needed a multibillion-dollar bailout.

But nobody wanted to annoy the TWU. It seems likely that the arbitrators were brought in to insulate the pols from public anger. Just two weeks ago, Paterson maintained this fiction, saying that, though “we don’t have the money,” the arbitrators “probably made the correct ruling technically.”

And the MTA wasn’t exactly careful, on behalf of the taxpayers, to assure a pristine process. It was almost unbelievably outrageous, as we learned long after the fact, that the “indepen dent” arbitrator on the three-man panel — former Deputy Mayor John Zuccotti, who represented the public — agreed, with the MTA’s support, to fork over his $116,000 “fee” to a TWU-controlled charity.

That is, the MTA used a supposedly independent process to wash a payment back to its “adversary” in the arbitration.

Plus, the agency was careless with the taxpayers’ money. The TWU, which is paying half of Zuccotti’s bill, obviously had no interest in keeping his $900-an- hour fee down or holding a bid to find someone to do it cheaper. The more Zuccotti cost, the bigger a donation the TWU would get. This didn’t seem to bother the MTA much, either, however.

Was then-MTA chief Elliot Sander living in such a public-sector bubble that he didn’t realize how this bonus would look? Or that it also risked helping to invalidate the arbitration process, wasting millions in legal fees?

And did someone in Albany realize the potential for outrage here — and figure it was a piece of handy future scapegoating?

Walder hasn’t paid Zuccotti’s fee, and is looking into it. But while the fee is a point of useful outrage (perhaps too useful), it’s really only a part of the problem: that a bizarre, expensive and lengthy process led, potentially, to $300 million a year in extra costs for the taxpayers — and we still have only a fuzzy idea of what happened.

For example: Why did the MTA at the last minute drop its heftiest request for a cost-saving work-rule change that could cost more than Zuccotti’s fee? And, more generally: Is the MTA’s supposed insulation from raw politics a fiction?

Walder is uninterested in a full-scale probe, saying that the goal should be “what can we do from where we are today.” He adds, somewhat contradictorily, that he isn’t working from “a blank sheet of paper.”

But we only get one chance every three years to make a big difference on TWU costs. It isn’t right for the MTA and Albany, institutionally, to think that they can squander that chance on backroom deals.

But an investigation would do more than deter future shenanigans: It would show why the MTA’s labor strategy should be a public one, not a private one.

The MTA should make a habit of laying out exactly what it needs from its workforce, including specific work-rule changes that would mean targeted cost savings as well as changes to future workers’ pensions. Then, it could lay out what it can do in negotiations on its own — and what it needs Albany’s help on (like pensions).

If politicians like Assemblyman Richard Brodsky who rail so much about public-authority opacity really want to fix things, this, too, is what they would agitate for.

Labor relations go to the heart of the MTA’s credibility, because people think that the agency is a swamp where Albany can give favors paid by taxpayers in secret.

And with the MTA, the past is never really past — because we are just starting to pay.

Nicole Gelinas, contributing editor to the Manhattan Insti tute’s City Journal, is author of “After The Fall: Saving Capital ism From Wall Street — and Washington.”